Paper
Large Components and Trees of Random Mappings
Authors
Ljuben Mutafchiev, Steven Finch
Abstract
Let be the set of all mappings , where . The corresponding graph of , called a functional digraph, is a union of disjoint connected components. Each component is a directed cycle of rooted labeled trees. We assume that each is chosen uniformly at random from the set . The components and trees of are distinguished by their size. In this paper, we compute the limiting conditional probability () that a vertex from the largest component of the random graph , chosen uniformly at random from , belongs to its -th largest tree, where is a fixed integer. This limit can be also viewed as an approximation of the probability that the -th largest tree of is a subgraph of its largest component, which is a solution of a problem suggested by Mutafchiev and Finch (2024).